Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(10,11)(12,16)(17,18,21,24)(19,20,23,22)(27,31)(30,32);(1,2,4,6)(3,5,7,8)(10,11)(14,15)(17,19,21,23)(18,22,24,20)(26,29)(30,32);(3,7)(5,8)(9,10,13,11)(12,14,16,15)(17,20,21,22)(18,23,24,19)(25,26,28,29)(27,30,31,32);(1,3,4,7)(2,5,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,27,28,31)(26,30,29,32);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,19)(18,20)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=i,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1814411);